"The Chaser" employs what's called situational irony. This is a situation where actions have an effect that is the exact opposite of what was intended. Alan wants to buy a love potion but will almost certainly end up returning to buy poison to kill his future wife when he gets tired of her constant, unwavering attention.
Another example of irony comes in the huge disparity in cost between the love potion and the poison, the "chaser" of the title. One would think that love would be much more precious, more valuable. And yet the love potion costs only a dollar, whereas the poison will set Alan back a cool five thousand dollars.
Then we have an example of verbal irony. When the old man says "Please a customer with one article, and he will come back when he needs another," he sounds ever so kindly and helpful. Yet in actual fact he's subtly hinting at the sinister nature of his business.
Finally, there's dramatic irony, where we know something one of the characters doesn't. We sense that when the old man tells Alan how his future wife "will never allow you to tire, to sit in a draught, to neglect your food" that this is something that he'll soon grow weary of. Hence the old man is able to assert, with complete confidence, that his customers will always come back to buy some "life-cleaner."